Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models

Authors: Boris Kramer, Benjamin Peherstorfer, and Karen WillcoxAuthors Info & Affiliations

http://doi.org/10.1137/16M1088958

Tools

Abstract

We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive, and accurately inferring parameters can be expensive, too. We address both problems by proposing an offline-online strategy for controlling systems with time-varying parameters. During the offline phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reflects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider two numerical test problems in the form of partial differential equations: a convection-diffusion system, and a model for flow through a porous medium. We demonstrate on those models that the proposed method successfully stabilizes the system model in the presence of process noise.

Keywords

MSC codes

Get full access to this article

View all available purchase options and get full access to this article.

References

1.

I. Akhtar, J. Borggaard, J. A. Burns, H. Imtiaz, and L. Zietsman, Using functional gains for effective sensor location in flow control: A reduced-order modelling approach, J. Fluid Mech., 781 (2015), pp. 622--656, http://doi.org/10.1017/jfm.2015.509.

2.

A. Alla and M. Falcone, An adaptive POD approximation method for the control of advection-diffusion equations, in Control and Optimization with PDE Constraints, K. Bredies, C. Clason, K. Kunisch, and G. von Winckel, eds., Springer, Basel, 2013, pp. 1--17.

3.

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005.

4.

J. Atwell, J. Borggaard, and B. King, Reduced order controllers for Burgers' equation with a nonlinear observer, Appl. Math. Comput. Sci., 11 (2001), pp. 1311--1330.

5.

G. Becker and A. Packard, Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback, Systems Control Lett., 23 (1994), pp. 205--215.

6.

P. Benner, M. Heinkenschloss, J. Saak, and H. K. Weichelt, An inexact low-rank Newton--ADI method for large-scale algebraic Riccati equations, Appl. Numer. Math., 108 (2016), pp. 125--142.

7.

P. Benner, H. Mena, and J. Saak, On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations, Electron. Trans. Numer. Anal., 29 (2008), pp. 136--149.

8.

P. Benner and J. Saak, Numerical Solution of Large and Sparse Continuous Time Algebraic Matrix Riccati and Lyapunov Equations: A State of the Art Survey, Preprint MPIMD/13-07, Max Planck Institute Magdeburg, 2013.

9.

J. Borggaard and M. Stoyanov, An efficient long-time integrator for Chandrasekhar equations, in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 3983--3988, http://doi.org/10.1109/CDC.2008.4738965.

10.

B. Brunton, S. Brunton, J. Proctor, and J. Kutz, Sparse sensor placement optimization for classification, SIAM J. Appl. Math., 76 (2016), pp. 2099--2122.

11.

S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. USA Acad. Sci., 113 (2016), pp. 3932--3937, http://doi.org/10.1073/pnas.1517384113.

12.

S. L. Brunton, J. H. Tu, I. Bright, and J. N. Kutz, Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1716--1732.

13.

M. Döhler and L. Mevel, Fast multi-order computation of system matrices in subspace-based system identification, Control Eng. Practice, 20 (2012), pp. 882--894.

14.

D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289--1306.

15.

Z. Drmac, S. Gugercin, and C. Beattie, Quadrature-based vector fitting for discretized $\mathcal{H}_2$ approximation, SIAM J. Sci. Comput., 37 (2015), pp. A625--A652.

16.

F. Feitzinger, T. Hylla, and E. W. Sachs, Inexact Kleinman--Newton method for Riccati equations, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 272--288.

17.

F. Guéniat, L. Mathelin, and M. Hussaini, A statistical learning strategy for closed-loop control of fluid flows, Theor. Computational Fluid Dyn., 30 (2016), pp. 497--510, http://doi.org/10.1007/s00162-016-0392-y.

18.

B. Gustavsen and A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Trans. Power Delivery, 14 (1999), pp. 1052--1061.

19.

A. HervÃ©, D. Sipp, P. J. Schmid, and M. Samuelides, A physics-based approach to flow control using system identification, J. Fluid Mech., 702 (2012), pp. 26--58, http://doi.org/10.1017/jfm.2012.112.

20.

J. G. In͂igo, D. Sipp, and P. J. Schmid, A dynamic observer to capture and control perturbation energy in noise amplifiers, J. Fluid Mech., 758 (2014), pp. 728--753.

21.

A. Ionita and A. Antoulas, Data-driven parametrized model reduction in the Loewner framework, SIAM J. Sci. Comput., 36 (2014), pp. A984--A1007.

22.

J.-N. Juang and R. Pappa, An eigensystem realization algorithm for modal parameter identification and model reduction, J. Guidance, Control, and Dynamics, 8 (1985), pp. 620--627.

23.

B. Kramer, P. Grover, P. Boufounos, S. Nabi, and M. Benosman, Sparse sensing and DMD based identification of flow regimes and bifurcations in complex flows, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1164--1196, http://doi.org/10.1137/15M104565X.

24.

B. Kramer and S. Gugercin, Tangential interpolation-based eigensystem realization algorithm for MIMO systems, Math. Comput. Model. Dynamical Syst., 22 (2016), pp. 282--306, http://doi.org/10.1080/13873954.2016.1198389.

25.

S.-Y. Kung, A new identification and model reduction algorithm via singular value decomposition, in Proc. 12th Asilomar Conf. Circuits, Syst. Comput., Pacific Grove, CA, 1978, pp. 705--714.

26.

K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), pp. 345--371.

27.

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972.

28.

C. Lee, J. Kim, D. Babcock, and R. Goodman, Application of neural networks to turbulence control for drag reduction, Phys. Fluids, 9 (1997), pp. 1740--1747, http://doi.org/10.1063/1.869290.

29.

Z. Ma, S. Ahuja, and C. Rowley, Reduced-order models for control of fluids using the eigensystem realization algorithm, Theoret. Comput. Fluid Dyn., 25 (2011), pp. 233--247.

30.

L. Mathelin, M. Abbas-Turki, L. Pastur, and H. Abou-Kandil, Closed-loop fluid flow control using a low dimensional model, Math. Comput. Modelling, 52 (2010), pp. 1161--1168, http://doi.org/10.1016/j.mcm.2010.01.007.

31.

L. Mathelin, L. Pastur, and O. Le Ma\^\itre, A compressed-sensing approach for closed-loop optimal control of nonlinear systems, Theoret. Comput. Fluid Dyn., 26 (2012), pp. 319--337.

32.

A. Mayo and A. Antoulas, A framework for the solution of the generalized realization problem, Linear Algebra Appl., 425 (2007), pp. 634--662.

33.

S. Nicaise, S. Stingelin, and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), pp. 555--573.

34.

B. Peherstorfer and K. Willcox, Data-driven operator inference for nonintrusive projection-based model reduction, Comput. Methods Appl. Mech. Engrg., 306 (2016), pp. 196--215.

35.

C. Poussot-Vassal and D. Sipp, Parametric reduced order dynamical model construction of a fluid flow control problem, IFAC-PapersOnLine, 48 (2015), pp. 133--138.

36.

J. L. Proctor, S. L. Brunton, and J. N. Kutz, Dynamic mode decomposition with control, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 142--161.

37.

C. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), pp. 115--127.

38.

E. Sachs and S. Volkwein, Pod-galerkin approximations in pde-constrained optimization, GAMM-Mitteilungen, 33 (2010), pp. 194--208.

39.

S. Sargsyan, S. L. Brunton, and J. N. Kutz, Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries, Phys. Rev. E, 92 (2015), 033304.

40.

P. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), pp. 5--28.

41.

P. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), pp. 5--28, http://doi.org/10.1017/S0022112010001217.

42.

V. Simoncini, D. B. Szyld, and M. Monsalve, On two numerical methods for the solution of large-scale algebraic Riccati equations, IMA J. Numer. Anal., 34 (2013), pp. 904--920.

43.

J. R. Singler and B. Kramer, A POD projection method for large-scale algebraic Riccati equations, Numer. Algebra, Control Optim., 6 (2016), pp. 413--435, http://doi.org/10.3934/naco.2016018.

44.

L. Sirovich, Turbulence and the dynamics of coherent structures. I. Coherent structures, Quart. Appl. Math., 45 (1987), pp. 561--571.

45.

G. Tissot, L. Cordier, and B. R. Noack, Feedback stabilization of an oscillating vertical cylinder by POD reduced-order model, in J. Physics: Conference Series, 574 (2015), 012137.

46.

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1 (2014), pp. 391--421.

47.

S. Volkwein, Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling, lecture notes, University of Konstanz, (2013), http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/scripts.php.

48.

F. Wang and V. Balakrishnan, Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems, IEEE Trans. Automat. Control, 47 (2002), pp. 720--734.

49.

X. Wu, V. Kumar, J. R. Quinlan, J. Ghosh, Q. Yang, H. Motoda, G. J. McLachlan, A. Ng, B. Liu, S. Y. Philip, Z. Zhou, M. Steinbach, D. Hand, and D. Steinberg, Top 10 algorithms in data mining, Knowledge Informat. Syst., 14 (2008), pp. 1--37.